Complex or imaginary shapes?

Algebra Level 5

The absolute value of a complex number would be the distance between the complex number to the origin \((0,0)\) in the complex plane. Or in other words, \[|a+bi| =\sqrt { { a }^{ 2 }+{ b }^{ 2 } } \] So for \(|z|=n\), the possible values of \(z\) would form a circle of radius \(n\) centered at the origin on the complex plane.

Now, supposing I create a new function: \(\ddagger a+bi\ddagger =|a|+|b|\)

And all of the possible values of \(z\) in \(\ddagger z\ddagger =2015\) forms a shape of area \(A\) on the complex plane.

Find \(\left\lfloor A \right\rfloor \)

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